[132] On the Dispersion of Light, as explained by the Hypothesis of Finite Intervals. Camb. Trans. vol. vi. p. 153.
[133] Investigation of the Equation to Fresnel’s Wave Surface, ib. p. 85. See also, in the same volume, Mathematical Considerations on the Problem of the Rainbow, showing it to belong to Physical Optics, by R. Potter, Esq., of Queen’s College.
We may be permitted to add, as a reflection obviously suggested by these facts, that the cause of the progress of science is incalculably benefited by the existence of a body of men, trained and stimulated to the study of the higher mathematics, such as exist in the British universities, who are thus prepared, when an abstruse and sublime theory comes before the world with all the characters of truth, to appreciate its evidence, to take steady hold of its principles, to pursue its calculations, and thus to convert into a portion of the permanent treasure and inheritance of the civilized world, discoveries which might otherwise expire with the great geniuses who produced them, and be lost for ages, as, in former times, great scientific discoveries have sometimes been.
The reader who is acquainted with the history of recent optical discovery, will see that we have omitted much which has justly excited admiration; as, for example, the phenomena produced by glass under heat or pressure, noticed by MM. Lobeck, and Biot, and Brewster, and many most curious properties of particular minerals. We have omitted, too, all notice of the phenomena and laws of the absorption of light, which hitherto stand unconnected with the theory. But in this we have not materially deviated from our main design; for our end, in what we have done, has been to trace the advances of Optics [131] towards perfection as a theory; and this task we have now nearly executed as far as our abilities allow.
We have been desirous of showing that the type of this progress, in the histories of the two great sciences, Physical Astronomy and Physical Optics, is the same. In both we have many Laws of Phenomena detected and accumulated by acute and inventive men; we have Preludial guesses which touch the true theory, but which remain for a time imperfect, undeveloped, unconfirmed: finally we have the Epoch when this true theory, clearly apprehended by great philosophical geniuses, is recommended by its fully explaining what it was first meant to explain, and confirmed by its explaining what it was not meant to explain. We have then its Progress struggling for a little while with adverse prepossessions and difficulties; finally overcoming all these, and moving onwards, while its triumphal procession is joined by all the younger and more vigorous men of science.
It would, perhaps, be too fanciful to attempt to establish a parallelism between the prominent persons who figure in these two histories. If we were to do this, we must consider Huyghens and Hooke as standing in the place of Copernicus, since, like him, they announced the true theory, but left it to a future age to give it development and mechanical confirmation; Malus and Brewster, grouping them together, correspond to Tycho Brahe and Kepler, laborious in accumulating observations, inventive and happy in discovering laws of phenomena; and Young and Fresnel combined, make up the Newton of optical science.
[2nd Ed.] [In the Report on Physical Optics, (Brit. Ass. Reports, 1834,) by Prof. Lloyd, the progress of the mathematical theory after Fresnel’s labors is stated more distinctly than I have stated it, to the following effect. Ampère, in 1828, proved Fresnel’s mathematical results directly, which Fresnel had only proved indirectly, and derived from his proof Fresnel’s beautiful geometrical construction. Prof. Mac Cullagh not long after gave a concise demonstration of the same theorem, and of the other principal points of Fresnel’s theory. He represents the elastic force by means of an ellipsoid whose axes are inversely proportional to those of Fresnel’s generating ellipsoid, and deduces Fresnel’s construction geometrically. In the third Supplement to his Essay on the Theory of Systems of Rays (Trans. R. I. Acad. vol. xvii.), Sir W. Hamilton has presented that portion of Fresnel’s theory which relates to the fundamental problem of the determination of the velocity and polarization of a plane wave, in a very elegant and analytical form. This he does by means of what he calls the [132] characteristic function of the optical system to which the problem belongs. From this function is deduced the surface of wave-slowness of the medium; and by means of this surface, the direction of the rays refracted into the medium. From this construction also Sir W. Hamilton was led to the anticipation of conical refraction, mentioned [above].
The investigations of MM. Cauchy and Lamé refer to the laws by which the particles of the ether act upon each other and upon the particles of other bodies;—a field of speculation which appears to me not yet ripe for the final operations of the analyst.
Among the mathematicians who have supplied defects in Fresnel’s reasoning on this subject, I may mention Mr. Tovey, who treated it in several papers in the Philosophical Magazine (1837–40). Mr. Tovey’s early death must be deemed a loss to mathematical science.
Besides investigating the motion of symmetrical systems of particles which may be supposed to correspond to biaxal crystals, Mr. Tovey considered the case of unsymmetrical systems, and found that the undulations propagated would, in the general case, be elliptical; and that in a particular case, circular undulations would take place, such as are propagated along the axis of quartz. It appears to me, however, that he has not given a definite meaning to those limitations of his general hypothesis which conduct him to this result. Perhaps if the hypothetical conditions of this result were traced into detail, they would be found to reside in a screw-like arrangement of the elementary particles, in some degree such as crystals of quartz themselves exhibit in their forms, when they have plagihedral faces at both ends.